Finite Group Databasebeta

← #360671

Subgroups — #360671

Order 512.

1348 conjugacy classes of subgroups.

The subgroup lattice has 589 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2314C211yesmaximal, characteristic
2C2×#1846C211yesmaximal, characteristic, complemented
3C2×#1836C211yesmaximal, characteristic, complemented
4C2×#1750C211yesmaximal, characteristic
5#14340C211yesmaximal, characteristic
6#11096C212yesmaximal
7#11095C212yesmaximal
8#10908C212yesmaximal, complemented
9#10907C212yesmaximal, complemented
10#10843C211yesmaximal, characteristic, complemented
11#10841C211yesmaximal, characteristic, complemented
12C22×#117C2211yescharacteristic
13C22×#107C2212yes—
14C22×#106C2212yes—
15C22×C22⋊C8C2211yescharacteristic
16C22×C22⋊C8C2211yescharacteristic
17C22×#97C2211yescharacteristic
18C22×C4×C8C2211yescharacteristic
19#2314—24no—
20#2314—24no—
21C2×#388—22no—
22C2×#387—22no—
23C2×#384C2211yescharacteristic
24C2×#384C2211yescharacteristic
25C2×#382C2211yescharacteristic
26C2×#382C2211yescharacteristic
27C2×#380C2212yes—
28C2×#380C2212yes—
29C2×C4×C4⋊C4C2212yes—
30#1846—24no—
31#1836—24no—
32#1750C418yescomplemented
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#336C2211yescharacteristic
40C2×#336C2211yescharacteristic
41#1441—22no—
42#1439—22no—
43C4⋊#95C2212yes—
44C24×C4C2311yescharacteristic
45C23×C8C2311yescharacteristic
46C23×C8C2311yescharacteristic
47C23×C8C2311yescharacteristic
48C23×C8C2311yescharacteristic
49C2×#117—22no—
50C2×#117—22no—
51C2×#117—22no—
52C2×#117—22no—
53C2×#117C2×C414yes—
54C2×#117Q814yes—
55C22×C4⋊C4—22no—
56C22×C4⋊C4—22no—
57C22×C4⋊C4D412yes—
58C22×C4⋊C4C2312yes—
59C22×C4⋊C4C2312yes—
60C22×C4⋊C4C2312yes—
61C22×C4⋊C4C2312yes—
62C22×C22⋊C4—22no—
63C22×C22⋊C4—22no—
64C22×C22⋊C4D412yes—
65C22×C42—22no—
66C22×C42C2311yescharacteristic
67C22×C42C2311yescharacteristic
68C2×#107—24no—
69C2×#107—24no—
70C2×#107—42no—
71C2×#106—24no—
72C2×#106—24no—
73C2×#106—42no—
74C2×C22⋊C8—22no—
75C2×C22⋊C8—22no—
76C2×C22⋊C8—22no—
77C2×C22⋊C8—22no—
78C2×C22⋊C8C2×C414yes—
79C2×C22⋊C8C2×C414yes—
80C2×#97—22no—
81C2×#97—22no—
82C2×#97C2×C414yes—
83C2×C4×C8—22no—
84C2×C4×C8—22no—
85C2×C4×C8C2×C414yes—
86#388—216no—
87#387—216no—
88#384—22no—
89#384—22no—
90#384—22no—
91#384D414yes—
92#382—22no—
93#382—22no—
94#382—22no—
95#382D414yes—
96#380—24no—
97#380—24no—
98#380—42no—
99#380—42no—
100C4×C4⋊C4—24no—
101C4×C4⋊C4—24no—
102C2×#95—22no—
103C2×#95—22no—
104C2×#95—22no—
105C2×#95—22no—
106C2×#95—22no—
107C2×#95—22no—
108C2×#95—22no—
109#340—24no—
110#340—24no—
111#340—24no—
112#340—24no—
113#340—42no—
114#340—42no—
115#336C2×C414yes—
116#336C2×C414yes—
117C25C2×D411yescharacteristic
118C23×C4—21no—
119C23×C4—21no—
120C23×C4—21no—
121C23×C4—21no—
122C23×C4—21no—
123C23×C4—21no—
124C23×C4—21no—
125C23×C4—21no—
126C23×C4—21no—
127C23×C4—21no—
128C23×C4—21no—
129C23×C4—21no—
130C23×C4—22no—
131C23×C4—22no—
132C23×C4—22no—
133C23×C4—22no—
134C23×C4C22×C412yes—
135C23×C4C2×D412yes—
136C23×C4C2×D411yescharacteristic
137C23×C4C2×D412yes—
138C23×C4C2×Q812yes—
139C23×C4C2411yescharacteristic
140C22×C8—21no—
141C22×C8—21no—
142C22×C8—21no—
143C22×C8—21no—
144C22×C8—21no—
145C22×C8—21no—
146C22×C8—21no—
147C22×C8—21no—
148C22×C8—21no—
149C22×C8—21no—
150C22×C8—21no—
151C22×C8—21no—
152C22×C8—21no—
153C22×C8—21no—
154C22×C8—21no—
155C22×C8—21no—
156C22×C8—21no—
157C22×C8—21no—
158C22×C8—21no—
159C22×C8—21no—
160C22×C8C22×C412yes—
161C22×C8C22×C412yes—
162C22×C8C22×C412yes—
163C22×C8C22×C412yes—
164C22×C8#4112yes—
165C22×C8#4112yes—
166C22×C8#4112yes—
167C22×C8#4112yes—
168#117—28no—
169#117—28no—
170#117—28no—
171#117—28no—
172C2×C4⋊C4—24no—
173C2×C4⋊C4—24no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—28no—
197C2×C4⋊C4—44no—
198C2×C4⋊C4—82no—
199C2×C4⋊C4—82no—
200C2×C4⋊C4—82no—
201C2×C4⋊C4—82no—
202C2×C4⋊C4C4⋊C418yes—
203C2×C4⋊C4C2×D414yes—
204C2×C4⋊C4C2×D414yes—
205C2×C22⋊C4—24no—
206C2×C22⋊C4—24no—
207C2×C22⋊C4—28no—
208C2×C22⋊C4—44no—
209C2×C22⋊C4—82no—
210C2×C22⋊C4—82no—
211C2×C22⋊C4—82no—
212C2×C22⋊C4—82no—
213C2×C22⋊C4C4⋊C418yes—
214C2×C42—21no—
215C2×C42—21no—
216C2×C42—21no—
217C2×C42—21no—
218C2×C42—28no—
219C2×C42—82no—
220C2×C42—82no—
221C2×C42C22×C412yes—
222C2×C42C22×C412yes—
223C2×C42C2×D412yes—
224C2×C42C2×D412yes—
225C2×C42C2×Q812yes—
226C2×C42C2×Q812yes—
227C2×C42#4112yes—
228C2×C42#4112yes—
229#107—48no—
230#106—48no—
231C22⋊C8—28no—
232C22⋊C8—28no—
233#97—28no—
234C4×C8—28no—
235#95—28no—
236#95—28no—
237#95—28no—
238#95—44no—
239#95—44no—
240#95—82no—
241#95—82no—
242C24—21no—
243C24—21no—
244C24—21no—
245C24—21no—
246C24—21no—
247C24—21no—
248C24—22no—
249C24—22no—
250C24—22no—
251C24—22no—
252C24C2×C4⋊C412yes—
253C24C22×D411yescharacteristic
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—21no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—24no—
283C22×C4—24no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4—41no—
292C22×C4—41no—
293C22×C4—41no—
294C22×C4—41no—
295C22×C4—41no—
296C22×C4—41no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4—41no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4C2×C4⋊C414yes—
327C22×C4C2×C4⋊C412yes—
328C22×C4C2×C4⋊C414yes—
329C22×C4#12014yes—
330C22×C4#12314yes—
331C22×C4C23×C411yescharacteristic
332C22×C4C22×D411yescharacteristic
333C22×C4C22×D411yescharacteristic
334C22×C4C22×Q811yescharacteristic
335C22×C4C2×#4111yescharacteristic
336C22×C4C2×#4111yescharacteristic
337C22×C4C2×#4111yescharacteristic
338C22×C4C2×#4111yescharacteristic
339C22×C4C2×#4111yescharacteristic
340C22×C4C2×#4111yescharacteristic
341C22×C4H(5,𝔽2)11yescharacteristic
342C22×C4H(5,𝔽2)11yescharacteristic
343C22×C42−1+411yescharacteristic
344C22×C42−1+411yescharacteristic
345C2×C8—22no—
346C2×C8—22no—
347C2×C8—22no—
348C2×C8—22no—
349C2×C8—22no—
350C2×C8—22no—
351C2×C8—22no—
352C2×C8—22no—
353C2×C8—22no—
354C2×C8—22no—
355C2×C8—22no—
356C2×C8—22no—
357C2×C8—22no—
358C2×C8—22no—
359C2×C8—22no—
360C2×C8—22no—
361C2×C8—22no—
362C2×C8—22no—
363C2×C8—41no—
364C2×C8—41no—
365C2×C8—41no—
366C2×C8—41no—
367C2×C8—41no—
368C2×C8—41no—
369C2×C8—41no—
370C2×C8—41no—
371C2×C8#11714yes—
372C2×C8#11714yes—
373C4⋊C4—216no—
374C4⋊C4—24no—
375C4⋊C4—24no—
376C4⋊C4—24no—
377C4⋊C4—24no—
378C4⋊C4—42no—
379C4⋊C4—42no—
380C4⋊C4—42no—
381C4⋊C4—42no—
382C4⋊C4—42no—
383C4⋊C4—42no—
384C4⋊C4—42no—
385C4⋊C4—42no—
386C4⋊C4—42no—
387C4⋊C4—42no—
388C4⋊C4—42no—
389C4⋊C4—42no—
390C4⋊C4—88no—
391C4⋊C4—88no—
392C22⋊C4—216no—
393C22⋊C4—88no—
394C22⋊C4—88no—
395C42—22no—
396C42—22no—
397C42—22no—
398C42—22no—
399C42—88no—
400C42C4×D414yes—
401C42C4×D414yes—
402C42C22⋊Q814yes—
403C42C22⋊Q814yes—
404C23—22no—
405C23—24no—
406C23—24no—
407C23—22no—
408C23—22no—
409C23—42no—
410C23—42no—
411C23—42no—
412C23—41no—
413C23—41no—
414C23—42no—
415C23—41no—
416C23—41no—
417C23—41no—
418C23—41no—
419C23—42no—
420C23—42no—
421C23—42no—
422C23—42no—
423C23—41no—
424C23—41no—
425C23—42no—
426C23—42no—
427C23C22×C4⋊C411yescharacteristic
428C23C2×#12011yescharacteristic
429C23C2×#12311yescharacteristic
430C23#53211yescharacteristic
431C23#53211yescharacteristic
432C23#53511yescharacteristic
433C23#53611yescharacteristic
434C23#57612yes—
435C23#57712yes—
436C23#57712yes—
437C23#57812yes—
438C2×C4—24no—
439C2×C4—24no—
440C2×C4—24no—
441C2×C4—24no—
442C2×C4—24no—
443C2×C4—24no—
444C2×C4—22no—
445C2×C4—24no—
446C2×C4—24no—
447C2×C4—22no—
448C2×C4—24no—
449C2×C4—22no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—44no—
467C2×C4—44no—
468C2×C4—44no—
469C2×C4—44no—
470C2×C4—44no—
471C2×C4—44no—
472C2×C4—82no—
473C2×C4—82no—
474C2×C4—82no—
475C2×C4—82no—
476C2×C4—82no—
477C2×C4—82no—
478C2×C4—82no—
479C2×C4—82no—
480C2×C4—82no—
481C2×C4—82no—
482C2×C4—82no—
483C2×C4—82no—
484C2×C4—82no—
485C2×C4—82no—
486C2×C4—82no—
487C2×C4—82no—
488C2×C4—82no—
489C2×C4—82no—
490C2×C4—82no—
491C2×C4—82no—
492C2×C4—82no—
493C2×C4—82no—
494C2×C4#38518yes—
495C2×C4C2×#11711yescharacteristic
496C2×C4C2×#11711yescharacteristic
497C2×C4C2×C4×D411yescharacteristic
498C2×C4C2×C4×D411yescharacteristic
499C2×C4#52011yescharacteristic
500C2×C4#52011yescharacteristic
501C2×C4C2×C22⋊Q811yescharacteristic
502C2×C4C2×C22⋊Q811yescharacteristic
503C2×C4#53311yescharacteristic
504C2×C4#53311yescharacteristic
505C2×C4#54411yescharacteristic
506C2×C4#54411yescharacteristic
507C2×C4#54611yescharacteristic
508C2×C4#54611yescharacteristic
509C2×C4#54711yescharacteristic
510C2×C4#54711yescharacteristic
511C2×C4#54811yescharacteristic
512C2×C4#54811yescharacteristic
513C2×C4#55011yescharacteristic
514C2×C4#55011yescharacteristic
515C2×C4#55211yescharacteristic
516C2×C4#55211yescharacteristic
517C2×C4#55311yescharacteristic
518C2×C4#55311yescharacteristic
519C2×C4#55511yescharacteristic
520C2×C4#55511yescharacteristic
521C2×C4#55611yescharacteristic
522C2×C4#55611yescharacteristic
523C8—24no—
524C8—24no—
525C8—42no—
526C8—42no—
527C22—42no—
528C22—42no—
529C22—42no—
530C22—44no—
531C22—44no—
532C22—82no—
533C22—81no—
534C22—81no—
535C22—82no—
536C22—81no—
537C22—81no—
538C22#228511yescharacteristic
539C22C2×#38511yescharacteristic
540C22#231711yescharacteristic
541C22#243111yescharacteristic
542C22#243211yescharacteristic
543C22#249911yescharacteristic
544C22#250011yescharacteristic
545C22#291212yes—
546C22#291312yes—
547C22#301212yes—
548C22#301312yes—
549C22#309612yes—
550C22#309712yes—
551C22#317312yes—
552C22#317411yescharacteristic
553C22#317411yescharacteristic
554C22#317512yes—
555C22#317611yescharacteristic
556C22#317711yescharacteristic
557C22#322412yes—
558C22#322512yes—
559C22#324112yes—
560C22#324212yes—
561C4—28no—
562C4—42no—
563C4—44no—
564C4—84no—
565C4—84no—
566C4—162no—
567C4—162no—
568C4#235211yescharacteristic
569C4#235211yescharacteristic
570C4#235411yescharacteristic
571C4#235411yescharacteristic
572C4#243911yescharacteristic
573C4#243911yescharacteristic
574C4#244111yescharacteristic
575C4#244111yescharacteristic
576C2—82no—
577C2#1387911yescharacteristic
578C2#2082011yescharacteristic
579C2#2082111yescharacteristic
580C2#2084912yes—
581C2#2199212yes—
582C2#2561012yes—
583C2#2568211yescharacteristic
584C2#2568311yescharacteristic
585C2#2788612yes—
586C2#2790011yescharacteristic
587C2#2790111yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×#117 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C2×#1846C2semidirect
3. C2×#1836C2semidirect
8. #10908C2semidirect
9. #10907C2semidirect
10. #10843C2semidirect
11. #10841C2semidirect
32. #1750C4semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#23142
C2×#18462
C2×#18362
C2×#17502
#143402
#110962
#110952
#109082
#109072
#108432
#108412